Spectral Efficiency Analysis of Multi-Cell Massive MIMO Systems with Ricean Fading
aa r X i v : . [ c s . I T ] A ug Spectral Efficiency Analysis of Multi-CellMassive MIMO Systems with Ricean Fading
Pei Liu, Kai Luo, Da Chen, Tao Jiang, and Michail Matthaiou
Abstract
This paper investigates the spectral efficiency of multi-cell massive multiple-input multiple-outputsystems with Ricean fading that utilize the linear maximal-ratio combining detector. We firstly presentclosed-form expressions for the effective signal-to-interference-plus-noise ratio (SINR) with the leastsquares and minimum mean squared error (MMSE) estimation methods, respectively, which apply forany number of base-station antennas M and any Ricean K -factor. Also, the obtained results can beparticularized in Rayleigh fading conditions when the Ricean K -factor is equal to zero. In the following,novel exact asymptotic expressions of the effective SINR are derived in the high M and high Ricean K -factor regimes. The corresponding analysis shows that pilot contamination is removed by the MMSEestimator when we consider both infinite M and infinite Ricean K -factor, while the pilot contaminationphenomenon persists for the rest of cases. All the theoretical results are verified via Monte-Carlosimulations. Index Terms
Effective SINR, multi-cell massive multiple-input multiple-output, pilot contamination, Ricean fad-ing, spectral efficiency.
I. I
NTRODUCTION
In the design of future communication systems, spectral efficiency (SE) becomes one of thedominant targets [1]. Massive multiple-input multiple-output (MIMO) [2], where the base station(BS) is equipped with hundreds of antennas to serve tens of users in the same time-frequency
P. Liu, K. Luo, D. Chen, and T. Jiang are with the School of Electronic Information and Communications, Huazhong Universityof Science and Technology, Wuhan 430074, China (e-mail: { peil, kluo, chenda, taojiang } @hust.edu.cn).M. Matthaiou is with the Institute of Electronics, Communications and Information Technology (ECIT), Queen’s UniversityBelfast, BT3 9DT, Belfast, United Kingdom (e-mail: [email protected]). August 28, 2018 DRAFT resource block, has attracted substantial attention from academia and industry thanks to the thehigh SE gains provided by the massive array [3]–[5]. The fundamental problem of placing amassive number of antennas in a confined space, can be addressed by pushing the operatingfrequency in the milimeter wave band, where the wavelengths become inherently small. Werecall that mm-wave channels are typically modeled via the Ricean fading distribution due tothe presence of line-of-sight (LOS) or specular components [6]. Hence, the SE analysis in massiveMIMO systems with Ricean fading becomes a problem of practical relevance.We will now review the relevant state-of-the-art in the massive MIMO literature. We firstlynote the work of [3], which derived useful lower bounds on the uplink ergodic rate for differentclassical linear receivers and analyzed the SE performance over a single-cell massive MIMOsystem operating in Rayleigh fading for perfect channel state information (CSI) and imperfectCSI, respectively. The same authors extended part of the above results into multi-cell systems andalso obtained the corresponding lower bounds in [7]. The authors in [8] provided a closed-formexpression of the SE, explored the SE maximizing problem in multi-cell systems, and proposedthe optimal system parameters. Recently, [4] considered the achievable uplink rates based onboth the least squares (LS) and minimum mean squared error (MMSE) estimation methodsin multi-cell massive MIMO systems and investigated the pilot power allocation scheme tomaximize the minimum SE. As a general comment, most issues pertaining to the achievable SEof massive MIMO, massive MIMO systems in Rayleigh fading have been largely and extensivelycharacterized. Apart from that, some recent investigations have analyzed the SE performance inmassive MIMO systems based on Ricean fading. For example, for a single-cell environment,[6] examined the scaling law and obtained the approximate expressions of uplink ergodic rateby considering Ricean fading and both perfect CSI and imperfect CSI. Moreover, for multi-cellmassive MIMO systems, [9] obtained a closed-form approximation and [10] provided a lowerbound of the achievable uplink rate with imperfect CSI, respectively. Also, [11] considereda similar scenario as [9] and pursued an asymptotic analysis of the achievable rate. Recently,[12], [13] systematically investigated the SE performance with spatially correlated Ricean fadingchannels and obtained closed-form expressions of uplink/downlink SE for different channelestimation methods. However, it appears that those expressions are untraceable to gain a veryclear insight. Also, the corresponding asymptotic analysis in massive antenna regime is obtainedat the cost of the tougher conditions of the spatial covariance matrix and LOS component. Fromthe above discussion, it becomes apparent that a straightforward and exact theoretical analysis
August 28, 2018 DRAFT of massive MIMO systems with Ricean fading is missing from the open literature.In this paper, we firstly introduce a general analytic framework for investigating the achievableuplink SE with a linear maximal-ratio combining (MRC) detector. Then, we derive two exactclosed-form expressions of the effective signal-to-interference-plus-noise ratio (SINR) based onthe LS and MMSE estimation methods, respectively, which apply for any number of antennas M and any Ricean K -factor. These expressions are particularly tractable for admitting fast andeffective computation. When the Ricean K -factor is equal to zero, i.e., Rayleigh fading, ourresults are substantially simplified. Also, based on the proposed expressions, we investigate indetail the performance of the effective SINR precisely by making either the M or the Ricean K -factor becomes infinite. In both cases, it is shown that the effect of pilot contamination cannotbe removed for both LS and MMSE estimation methods; however, if both the M and Ricean K -factor continue to grow unbound, the pilot contamination issue is eliminated only for theMMSE estimation method. Finally, a set of Monte-Carlo simulations is conducted to validatethe the above mentioned analytical results. Notation:
Lower-case and upper-case boldface letters denote vectors and matrices, respectively; C M × N denotes the M × N complex space; A T , A † , and A − denote the transpose, the Hermitiantranspose, and the inverse of the matrix A , respectively; I M denotes an M × M identity matrixand M × N denotes an M × N zero matrix. The expectation operation is E {·} . A complexGaussian random vector x is denoted as x ∼ CN (¯ x , Σ ) , where the mean vector is ¯ x and thecovariance matrix is Σ , while k · k denotes the 2-norm of a vector. Finally, diag ( a ) denotes adiagonal matrix where the main diagonal entries are the elements of vector a .II. S YSTEM M ODEL
In this paper, we consider a typical uplink cellular communication system with L hexagonalcells. Each cell contains a BS and N single-antenna users. Each BS has a uniform linear arraywith M ( M ≫ N ) antennas. In the following, the time-division duplex (TDD) mode is adoptedand we assume that the BSs and users in this system are perfectly synchronized. For each channeluse, the M × received signal vector of the BS in cell j is given by y j = √ ρ u L X l =1 H jl x l + n j , (1)where x l denotes the N × vector containing the transmitted signals from all the users in cell l ,which satisfies E { x l } = N × and E n x l x † l o = I N , ρ u is the average transmitted power of eachuser, and n j ∈ C M × ∼ CN ( M × , I M ) is the additive white Gaussian noise (AWGN) vector. August 28, 2018 DRAFT
Also, H jl ∈ C M × N represents the channel matrix between the users in cell l and the BSin cell j , whose its n th column, h jln ∈ C M × , is the channel vector between the user n incell l and the BS in cell j . Here, the block-fading model [3] is utilized where the large-scalefading coefficients are kept fixed over lots of coherence time intervals and the small-scale fadingcoefficients remain fixed within a coherence time interval and change between any two adjacentcoherence time intervals. Moreover, the large-scale fading coefficients are assumed to be perfectlyknown at the BS side due to their slow-varying nature. We herein consider the Ricean fadingmodel in [9] where both a LOS path and a non LOS (NLOS) component exist in the channelsbetween the users and BS in the same cell, while only a NLOS component exists in the channelsbetween the users and BS in different cells. The above mentioned model is reasonable since aLOS component is more likely to kick in when the users and the BS are in the same cell. Hence, h jln can be rewritten as h jln = q K jn K jn +1 h jjn, LOS + q K jn +1 h jjn, NLOS , l = j, h jln, NLOS , l = j, (2)where K jn denotes the Ricean K -factor for the user n in cell j , h jln, NLOS ∼ CN ( M × , β jln I M ) ( ∀ l, n ) denotes the NLOS component between the user n in cell l and the BS in cell j , while β jln is the corresponding large-scale fading coefficient. Also, h jjn, LOS ∈ C M × denotes the LOSpart between the user n and the BS in cell j , whose m th entry [ h jjn, LOS ] m is given by [ h jjn, LOS ] m = β jjn e − i ( m − πdλ sin( θ jn ) , (3)where λ is the wavelength, d is the antenna spacing, θ jn ∈ [0 , π ) is the angle of arrival of the n th user in cell j , and i denotes imaginary unit. Based on the fact that K jn and θ jn can beobtained through a feedback link [14], we can also assume that the Ricean K -factor and LOSpath can be perfectly obtained by the BS and users.In practical communication systems, the BS side does not have perfect CSI and the channelneeds to be estimated at the BS. In TDD mode, uplink pilot training is adopted for obtainingthe estimated channel information before data transmission. The worst-case of pilot sequenceallocation is adopted here, where each cell’s users utilize the same set of pilot sequences [4],which we denote as Φ ∈ C τ × N . Also, τ is the length of the pilot sequence, which is larger August 28, 2018 DRAFT than or equal to N , and Φ satisfies Φ † Φ = I N . Then, the M × τ received pilot sequence signalmatrix at the BS in cell j is given by Y j, train = L X l =1 H jl ( Ω l + I N ) P l Φ † + N j , (4)where P l is the N × N diagonal matrix with the power of the pilot sequence sent by user n incell l is [ P l ] nn = ρ ln , N j ∈ C M × τ is the AWGN matrix with the independent and identicallydistributed zero-mean and unit-variance elements, and Ω l = diag([ K l , . . . , K ln , . . . , K lN ] T ) ∈ C N × N . Particularly, the reason for the pilot matrix P l Φ † multiplied by ( Ω l + I N ) , which issent by all users in cell j , is to estimate the NLOS component in a simplified manner; note thata similar method followed in [6], [9]. By removing the known LOS part in (4) and utilizing theLS and MMSE estimation methods [4], [15], h jjn, NLOS is estimated as ˆh LS jjn, NLOS = h jjn, NLOS + L X l = j s ρ ln ( K ln +1) ρ jn h jln, NLOS + ˜ n jn √ ρ jn , (5) ˆh MMSE jjn,
NLOS = χ jn ˆh LS jjn, NLOS , (6)respectively, where ˆh LS jjn, NLOS and ˆh MMSE jjn,
NLOS denote the estimators of h jjn, NLOS based on theLS and MMSE estimation methods, respectively, as well as, ˜ n jn , N j φ n . Also, φ n is the n thcolumn of Φ and χ jn , ρ jn β jjn ρ jn β jjn + L P l = j ρ ln ( K ln + 1) β jln + 1 . (7)For convenience, we denote the estimator of h jjn, NLOS as ˆh jjn, NLOS and the estimator of h jjn as ˆh jjn = s K jn K jn + 1 h jjn, LOS + s K jn + 1 ˆh jjn, NLOS , (8)respectively, for both the LS and MMSE estimation methods.After channel estimation, we use the standard linear detector MRC [4] to detect the receiveddata signal in (1). Hence, for the BS in cell j , (1) is separated into N streams by multiplying itwith the MRC detector, that is, r j = ˆ H † jj y j ∈ C N × , (9)where ˆ H jj , [ˆ h jj , . . . , ˆ h jjn , . . . , ˆ h jjN ] ∈ C M × N . Then, for the n th user in cell j , we have r jn = √ ρ u ˆ h † jjn h jjn x jn + √ ρ u X ( l,t ) =( j,n ) ˆ h † jjn h jlt x lt + ˆ h † jjn n j , (10)where r jn is the n th entry of r j . August 28, 2018 DRAFT
III. S
PECTRAL E FFICIENCY
In this section, we obtain the closed-form expressions of the effective SINR for both LSand MMSE estimation methods and, thereafter, aim to analyze the SINR performance with therespect of the number of BS antennas M and the Ricean K -factor, respectively. A. Closed-Form of SINR jn Since we want to obtain a computable expression of the achievable uplink SE and investigateit by a simple way, it is convenient to follow the methodology of [4] that assumes that the term E { ˆh † jjn h jjn } is known at the BS in cell j perfectly. Hence, (10) can be rewritten as r jn = √ ρ u E n ˆ h † jjn h jjn o x jn + √ ρ u X ( l,t ) =( j,n ) ˆ h † jjn h jlt x lt + √ ρ u n ˆ h † jjn h jjn − E n ˆ h † jjn h jjn oo x jn + ˆ h † jjn n j . (11)Then, by using the definition of the effective SINR in multi-cell massive MIMO systems as in[4, Eq. (18)], the achievable uplink SE of the n th user in cell j , in units of bit/s/Hz, is given by R jn = T − τT log (1 + SINR jn ) , (12)where T denotes the channel coherence time interval, in terms of the number of symbols, while τ symbols are utilized for channel estimation, and the SINR jn is defined asSINR jn , ρ u (cid:12)(cid:12)(cid:12) E n ˆ h † jjn h jjn o(cid:12)(cid:12)(cid:12) ρ u L P l =1 N P t =1 E (cid:26)(cid:12)(cid:12)(cid:12) ˆ h † jjn h jlt (cid:12)(cid:12)(cid:12) (cid:27) − ρ u (cid:12)(cid:12)(cid:12) E n ˆ h † jjn h jjn o(cid:12)(cid:12)(cid:12) + E (cid:26)(cid:13)(cid:13)(cid:13) ˆ h jjn (cid:13)(cid:13)(cid:13) (cid:27) . (13)The following theorem presents a new general framework for the closed-form expression ofSINR jn , which applies for both the LS and MMSE estimation methods. This constitutes a keycontribution of this paper. Theorem 1:
The exact SINR jn , for both the LS and MMSE estimation methods, can beanalytically evaluated as SINR jn = SINR LS jn , LS , SINR
MMSE jn , MMSE , (14) August 28, 2018 DRAFT whereSINR LS jn , M ρ jn ( K jn + 1) β jjn M ψ jn + ζ jn ϑ j + ρ jn K jn β jjn ς jn , (15)SINR MMSE jn , M ρ jn ( K jn + χ jn ) β jjn M χ jn ( K jn +1) ψ jn + ρ jn ( K jn + χ jn ) ( K jn +1) β jjn ϑ j + ρ jn K jn ( K jn +1) β jjn ς jn . (16)Also, ψ jn , ζ jn , ϑ j , and ς jn are denoted as ψ jn , L X l = j ρ ln ( K ln + 1) β jln , (17) ζ jn , L X c =1 ρ cn ( K cn + 1) β jcn + 1 , (18) ϑ j , L X l =1 N X t =1 β jlt + 1 ρ u , (19) ς jn , N X t = n K jt K jt + 1 φ nt M β jjt − N X t =1 K jt K jt + 1 β jjt , (20)respectively, where φ nt , sin (cid:0) Mπ (sin( θ jn ) − sin( θ jt )) (cid:1) sin (cid:0) π (sin( θ jn ) − sin( θ jt )) (cid:1) . (21) Proof:
See Appendix A.It is important to note that the expressions in
Theorem 1 can be easily evaluated since theyinvolve only the pilot sequence power, uplink data power, Ricean K -factor, and large-scalefading coefficients, for all cases of interest. Moreover, from Theorem 1 , we see that the obtainedeffective SINRs based on the LS and MMSE channel estimation methods with MRC detector aredifferent. Note that when K ln = 0 ( ∀ l, n ) , SINR jn reduces to the special case of Rayleigh fadingchannel. After performing some simplifications, for both LS and MMSE estimation methods,we have SINR Rayleigh ,jn = M ρ jn β jjn M L P l = j ρ ln β jln + (cid:18) L P c =1 ρ cn β jcn + 1 (cid:19) ϑ j . (22)Interestingly, (22) is the effective SINR in Rayleigh fading channels given by [4, Theorem 1].Note that Theorem 1 gives a universal formula for the SINR jn when Ricean fading is considered. August 28, 2018 DRAFT
B. Analysis of SINR jn Now, we consider the SINR jn limit when M grows without bound. To the best of ourknowledge, this result is also new. Corollary 1: If M → ∞ , the exact analytical expression of the SINR jn in (14) approaches to lim M →∞ SINR jn = SINR LS jn , LS , SINR
MMSE jn , MMSE , (23)where SINR LS jn , ρ jn ( K jn + 1) β jjn ψ jn , (24)SINR MMSE jn , ρ jn ( K jn + χ jn ) β jjn χ jn ( K jn + 1) ψ jn . (25) Proof:
The proof is completed by calculating the limit of (14) when M → ∞ . Corollary 1 indicates that if the number of users is kept fixed and the number of receiveantennas at the BS side is increased, then, the asymptotic SINR jn is saturated. Intuitively, this isdue to the pilot contamination since the other cells’ users adopt the same pilot sequence as theuser in the target cell j . It is also worth noting that, based on (24) and (25), if K ln = 0 , ∀ l, n ,the limit of SINR jn as M → ∞ for both LS and MMSE estimation methods is given bySINR Rayleigh ,jn = ρ jn β jjnL P l = j ρ ln β jln . (26)Again, it is important to note that (26) is the limit of effective SINR with imperfect CSI inRayleigh fading channels given by [4, Corollary 1]. Hence, (26) is a special case of (23) if thepower of the LOS part of the Ricean fading channel is equal to zero. To gain more insights,the exact SINR jn admits further simplifications in the large Ricean K -factor regime for both LSand MMSE estimation methods. Corollary 2:
If for any l and n , K ln = K → ∞ , (14) converges to lim K →∞ SINR jn = ] SINR LS jn , LS , ] SINR
MMSE jn , MMSE , (27) August 28, 2018 DRAFT where ] SINR LS jn and ] SINR
MMSE jn are defined as ] SINR LS jn , Mρ jn β jjn M L P l = j ρ ln β jln + L P c =1 ρ cn β jcn ϑ j + ρ jn β jjn ̺ jn , (28) ] SINR
MMSE jn , M β jjnL P l = j N P t =1 β jlt + ρ u + N P t = n φ nt M , (29)respectively. Also, ̺ jn is denoted as ̺ jn , N X t = n φ nt M β jjt − N X t =1 β jjt . (30) Proof:
The proof is completed by calculating the limit of (14) when K → ∞ .It is interesting to note from Corollary 2 that as K increases, the SINR based on LS andMMSE will approach two constant values, respectively. Note that (29) is unbounded as M → ∞ ,whereas (28) is bounded for the same conditions. In other words, in the high Ricean K -factorregime with infinite M , the pilot contamination effect will be completely eliminated for MMSEestimation, since this scheme accounts for the presence of a LOS component in (6). On the otherhand, pilot contamination effect cannot be be removed when using the LS estimation methodsince the LS estimation method regards the co-channel interference in (5) as just noise. Notethat, Corollary 2 reflects that massive MIMO systems have unlimited achievable uplink SE viadifferent estimation methods, the number of BS antennas, and Ricean K -factor, though in aslightly different way than in [16]. C. Numerical Results
In this section, we consider a hexagonal cellular network with a set of L cells and radius (fromcenter to vertex) ̟ meters where users are distributed uniformly in each cell. Also, leveragingthe large-scale fading model of [17], the large-scale fading coefficient between the user n in cell l and the BS in cell j , β jln , is given by β jln = v jln (cid:16) η jln η min (cid:17) α , (31)where v jln is a log-normal random variable with standard deviation ξ , α is the path loss exponent, η jln is the distance between the user n in cell l and the BS in cell j , and η min is the referencedistance. In our simulations, we choose L = 7 , N = 10 , ̟ = 500 m, ξ = 8 dB, α = 3 . , d = λ , August 28, 2018 DRAFT0
50 100 150 200 250 300 350 400 450 5005101520253035404550 Number of BS antennas M S u m a c h i e v ab l e up li n k SE R s u m ( B i t/ s / H z ) K =3dB, 6dB, and 10dB K =0(− ∞ dB) LS−AnalyticalLS−SimulationMMSE−AnalyticalMMSE−Simulation (a) Moderate-to-High M regime M S u m a c h i e v ab l e up li n k SE R s u m ( B i t/ s / H z ) K = 3dB, 6dB, and 10dB K = 3dB, 6dB, and 10dB K =0(− ∞ dB)LSMMSEAsymptotic (High M ) 10 (b) Whole M regimeFig. 1. The sum achievable uplink SE R sum as a function of the number of BS antennas M with ρ p = 30 dB, ρ u = 20 dB, aswell as, K = 0( −∞ dB ) , dB , dB , and dB, for the LS and MMSE estimation methods, respectively. and η min = 200 m, which follow the methodology of [6], [17]. Here, the observed cell is thecenter cell and call it cell 1, i.e., j = 1 . Also, the pilot sequence symbol and the data symbolare assumed to be modulated based on orthogonal frequency-division multiplexing (OFDM). Byconsidering the long term evolution standard, the channel coherence time interval is equal to196 OFDM symbols, i.e., T = 196 [3], [6]. Since we assume that the noise variance is 1, weconsider that each user has the same pilot sequence power denoted by ρ p which is equal to 30dB,and the uplink data power ρ u = 20 dB. For convenience, we assume all the channels between theBS and the users in same cell have the same Ricean K -factor, denoted by K , and τ = N = 10 OFDM symbols. Finally, all the simulation results are obtained by averaging 100 realizationsof all users’ large-scale fading coefficients in all cells over 100 independent small-scale fadingchannels for each realization of users’ large-scale fading coefficients.In the following, we assess the accuracy of the achievable uplink SE given by (12) for both LSand MMSE estimation methods, the closed-form expression given in
Theorem 1 , and the resultsin
Corollary 1 and
Corollary 2 . For comparison, we define the metric called “Sum achievableuplink SE” in target cell 1, which is given as R sum , N X n =1 R n . (32)Fig. 1(a) gives the analytical and Monte-Carlo simulated sum achievable uplink SE R sum withthe LS and MMSE estimation methods, respectively, in moderate-to-high M regime. Results August 28, 2018 DRAFT1 are shown for different Ricean K -factor, and pilot sequence power ρ p = 30 dB with uplink datapower ρ u = 20 dB. We see that in all cases the analytical curves (based on (14)) match preciselywith the simulated curves (based on (13)), which proves the validity of Theorem 1 . Moreover,for all cases, when M increases, R sum increases. Also, when the power of the LOS path becomeszero, R sum for both the LS and MMSE estimation methods are identical for both the analyticaland simulated curves, respectively, which not only shows the validity of the results (based on(22)) in previous literature [4], but also does prove that Theorem 1 can be applied into theRayleigh fading environment. Moreover, in Ricean fading conditions, the results in this figureshow that the MMSE performance is always better than the LS.Fig. 1(b) investigates the analytical results of sum achievable uplink SE R sum in the whole M regime under the same parameter setting in Fig. 1(a). When M → ∞ , we see that all R sum results tend to different constants, which match the asymptotic expression (based on (23)) fordifferent Ricean K -factor, respectively. In other words, it justifies the effectiveness of Corollary1 . Also, we note that, for the LS case with K = 0( −∞ dB ) , dB , dB , and dB, as well as,the MMSE case with K = 0( −∞ dB ) , the asymptotic results are identical. This phenomenon iscaused by the following two reasons. First, if all users’ Ricean K -factors are identical, (24) isuncorrelated with the Ricean K -factor. In other words, for the LS case, different Ricean K -factormeans the same asymptotic SE. Second, when Ricean K -factor is equal to zero, both the LSand MMSE cases have the same asymptotic SE since the current channel becomes the Rayleighfading channel. Hence, for both the LS and MMSE estimation methods, the pilot contaminationexists such that the R sum saturates even when M → ∞ .In Fig. 2, we investigate the impact of the Ricean K -factor on the sum achievable uplinkSE performance for M = 125 , , , and , with the LS and MMSE estimation methods,respectively. In this figure, the pilot sequence power is 30dB and the uplink data power is20dB. It shows that the analytical values and simulation values are almost indistinguishablefor both the LS and MMSE estimation methods, regardless of the number of BS antennas andRicean K -factors. Moreover, across the entire Ricean K -factor regime, for both LS and MMSEestimation methods, a larger M means larger R sum . Given the number of BS antennas M , the sumachievable uplink SE performance difference between the LS and and MMSE is distinguishable Since the match of the analytical results and simulation results has been examined in Fig. 1(a), for convenience, we onlyneed to examine the analytical results in Fig. 1(b).
August 28, 2018 DRAFT2 −10 0 10 20 30 401520253035404550556065 Ricean K −factor (dB) S u m a c h i e v ab l e up li n k SE R s u m ( B i t/ h z / s ) M =125 M =250 M =500 M =1000LS−AnalyticalLS−SimulationMMSE−AnalyticalMMSE−Simulation Asymptotic (High Ricean− K factor) Fig. 2. The sum achievable uplink SE R sum as the Ricean K -factor increases with ρ p = 30 dB, ρ u = 20 dB, as well as, M = 125 , , , and , for the LS and MMSE estimation methods, respectively. expect in the low Ricean K -factor regime since in this situation the Ricean fading channel tendsto become a Rayleigh fading channel. When the Ricean K -factor becomes infinite, the sumachievable uplink SE R sum approaches to different constant values, which match the asymptoticexpressions (based on (27) in Corollary 2 ) well, respectively. If also M → ∞ , it can be shownthat the pilot contamination is completely removed for the MMSE case.IV. C ONCLUSION
In this paper, a detailed statistical characterization of the SE for the muti-cell massive MIMOsystem with Ricean fading was presented. In order to evaluate the SE performance, we firstproposed two exact closed-form expressions for the effective SINR based on LS and MMSEestimation methods, respectively, which also can be adopted in Rayleigh fading. Then, weanalyzed the asymptotic properties of the effective SINR when the number of BS antennas M and the Ricean K -factor became infinite. It was shown that, when the Ricean K -factor becameinfinite or M → ∞ , the SINR performance for the LS and MMSE estimation methods wassaturated, which underlines the pilot contamination phenomenon. However, if both the Ricean K -factor and M grow asymptotically large, the pilot contamination phenomenon disappearedfor the MMSE estimation method, but persists for the LS estimation method. In this figure, the asymptotic case when Ricean K -factor approaches to −∞ dB ) has not been shown since the Rayleighfading case has been examined in Fig. 1. August 28, 2018 DRAFT3 A PPENDIX AP ROOF OF T HEOREM jk in (13), we define six terms A , (cid:12)(cid:12)(cid:12) E n ˆ h † jjn h jjn o(cid:12)(cid:12)(cid:12) , (33) B , E (cid:26)(cid:12)(cid:12)(cid:12) ˆ h † jjn h jjn (cid:12)(cid:12)(cid:12) (cid:27) , (34) C , E (cid:26)(cid:12)(cid:12)(cid:12) ˆ h † jjn h jjt (cid:12)(cid:12)(cid:12) (cid:27) , ( t = n ) , (35) D , E (cid:26)(cid:12)(cid:12)(cid:12) ˆ h † jjn h jln (cid:12)(cid:12)(cid:12) (cid:27) , ( l = j ) , (36) E , E (cid:26)(cid:12)(cid:12)(cid:12) ˆ h † jjn h jlt (cid:12)(cid:12)(cid:12) (cid:27) , ( l = j & t = n ) , (37) F , E (cid:26)(cid:13)(cid:13)(cid:13) ˆ h jjn (cid:13)(cid:13)(cid:13) (cid:27) . (38)Although ˆ h jjn has different expressions for the LS and MMSE estimation methods, thecorresponding proofs for SINR jn are similar. Hence, it is convenient to only study the caseof the LS estimation. • Calculate A : Substituting (3), (5) and (8) into (33) , after much algebraic manipulation, itcan be shown that A reduces to A = M β jjn . (39) • Calculate B : Substituting (2), (5), and (8) into (34), after some manipulations, it is easy toobtain B = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h † jjn L X l = j s ρ ln ( K ln +1) ρ jn h jln, NLOS + ˜ n jn √ ρ jn !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | {z } B K jn + 1 + E (cid:8) k h jjn k (cid:9)| {z } B , (40)where the closed-form expression of B can be obtained based on h jjn is uncorrelatedwith the rest of terms in B and the distribution of h jjn . Also, B can be obtained basedon the properties of non-central Wishart matrices [6, Eq. (123)]. After some algebraicmanipulations, we write B as follows B = M β jjn + M β jjn K jn ( K jn + 1) + M β jjn L P l = j ρ ln ( K ln + 1) β jln + 1 ρ jn ( K jn + 1) . (41) August 28, 2018 DRAFT4 • Calculate C : Since the ˆ h jjn is uncorrelated with h jjt when t = n , we substitute (2), (5),and (8) into C , as well as, utilize [6, Eq. (128)]. Then, clearly C = β jjn β jjt K jn K jt φ nt + M ( K jn + K jt ) + M ( K jn + 1)( K jt + 1) + M β jjt L P l = j ρ ln ( K ln + 1) β jln + 1 ρ jn ( K jn + 1) , (42)where φ nt has been defined as in (21). • Calculate D : Substituting (2), (5), and (8) into (36), and performing some basic simplifica-tions to obtain D = E (cid:8) | D | (cid:9) + E (cid:8) | D | (cid:9) , (43)where D and D is denoted as D , h jjn, LOS s K jn K jn +1 + h jjn, NLOS + L X c = j,l s ρ cn ( K cn +1) ρ jn h jcn, NLOS + ˜ n jn √ ρ jn !s K jn +1 ! † × h jln, NLOS , (44) D , s ρ ln ( K ln + 1) ρ jn ( K jn + 1) k h jln, NLOS k , (45)respectively. Based on the similar way for obtaining C , E (cid:8) | D | (cid:9) reduces to E (cid:8) | D | (cid:9) = M β jln L P c = j,l ρ cn ( K cn + 1) β jcn + 1 ρ jn ( K jn + 1) + M β jjn β jln . (46)By using the properties of Wishart matrices [18, Lemma 2.9], we obtain E (cid:8) | D | (cid:9) = ρ ln ( K ln + 1) ρ jn ( K jn + 1) M ( M + 1) β jln . (47)Therefore, substituting (46) and (47) into (43) and simplifying, we get D = M β jln ρ ln ( K ln + 1) ρ jn ( K jn + 1) + M β jjn β jln + M β jln L P c = j ρ cn ( K cn + 1) β jcn + 1 ρ jn ( K jn + 1) . (48) • Calculate E : Based on the similar way for obtaining B and C , E is given by E = M β jjn β jlt + M β jlt L P c = j ρ cn ( K cn + 1) β jcn + 1 ρ jn ( K jn + 1) . (49) • Calculate F : With the help of (3), (5), and (8), we get F = M β jjn + M L P l = j ρ ln ( K ln + 1) β jln + 1 ρ jn ( K jn + 1) . (50)Finally, substituting (39), (41), (42), and (48)-(50) into (13) and simplifying, the closed-formexpression for SINR LS jn is obtained. August 28, 2018 DRAFT5 R EFERENCES
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